Cognition, Practice, and Mathematical Oddities

[General,Projects] (11.05.08, 1:22 am)

One of my classes is cross listed with an undergraduate course, this is Nancy Nersessian‘s Cognition and Culture. One fun advantage to having a course with undergraduates is that they make a lot of interesting and occasionally profoundly brilliant observations. Not to say that us graduate students are incapable of insight, but we tend to be very bogged down by our own research objectives.

We have been discussing Jean Lave‘s book, Cognition in Practice, and came to a segment where Lave discusses how mathematics is a cultural artifact, but we view it as universal and supremely valuable. An example of this is that we “beam” the Pythagorean theorem into space, in hopes that, were the signal ever to be discovered by extraterrestrials, it would help communication because mathematics is a universal language, that transcends humanity. I didn’t find a source on this beaming precisely, but it seems like the sort of thing that people might do. Coming from a mathematical background, and moving into the complex and tricky field of cognitive science and cultural studies, I had very torn reactions to this conflict, and only realized how to articulate that reaction after the discussion ended. So, I present it here.

Mathematicians are extremely strange people. I don’t really identify as a mathematician anymore, but I still consider myself close to the culture, so I say this pridefully. The conclusions of mathematics are universal, and they are fundamental, but, and this is where things get difficult, these universal conclusions rely on premises. These premises are necessarily situational, and depend on other cultural factors. Furthermore, the practice of mathematics is also culturally relevant, and lots of mathematicians disagree, not on conclusions (a proof is a proof, after all), but on the relevance, importance, usefulness, and elegance of different practices of math. All of these terms are subjective, and while there are many common impressions of what elegance means, it is far from universal.

Generally issues regarding the practice of math applies to topics that are more sophisticated than the Pythagorean theorem. The Pythagorean theorem has to be universal because of its simplicity, elegance, and universality in almost all kinds of math that we use conventionally, right? Those aliens must use that kind of math too, right? Well, mostly. Even in this case, the situation is ambiguous, and that ambiguity arises from the premises under which the Pythagorean theorem is valid, namely: Euclidean geometry. If you are dealing with some other domain of planar geometry, (most notably, spherical or hyperbolic geometry), then the Pythagorean theorem breaks down. It has analogues (which are quite elegant, I might say), but the existence of these alternative types of geometries, and the ways in which the theorems are modified illustrates that our idyllic Euclidean world is not quite as simple or so complete as it first seemed. Space itself is non-Euclidean, according to both relativity and quantum mechanics. So perhaps the Pythagorean Theorem may actually have something to do with our experience as humans on Earth, and may not be quite so transcendent after all.

For real transcendence, we need Godel’s Incompleteness Theorem. That’ll do it for sure.

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